Genotypes, Relative Frequencies and Totals

I tell you that you and your friend will get a half a mango each.
(i.e., the ratio of your shares is 1:1).
Is it the same as saying that you will together get one mango?
(i.e., your share + your friend’s share = 1)

..
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With reference to genotype frequencies, do the following mean the same thing?

p²: 2pq: q²
andp² + 2pq + q² = 1

First, think about Mendelian ratios in Genetics

At a single locus with two alleles, what are the expected frequencies of genotypes after a cross between two F1 individuals (heterozygotes, Aa)?

In such a cross [Aa x Aa] the relative frequencies of genotypes in the F2 generation are expected to follow the proportions

freq AA : freq Aa : freq aa = p²: 2pq: q²

[because p =0.5 and q = 0.5, the ratio of these frequencies will always be 1 : 2 : 1]

The formulae (p², 2pq, q²) tell you how to predict or calculate the expected proportions of genotypes in the F2 generation, where p is the proportion of gamete A and q is the proportion of gamete a produced by each individual, Aa (50% in each case).

Regardless of how many pairs are crossed, there would always be 50% A gametes and 50% a gametes in each cross.

It is true, but only trivially important that the sum of these frequencies is 1:

Phenotypes: If one of the alleles A is dominant over the other a, the phenotypes would occur as

freq AA /Aa : freq aa = (p² + 2pq) : q²

Now, think about Hardy Weinberg ratios (Population Genetics)

After random mating among all genotypes (AA, Aa and aa) in a large population, the relative frequencies of genotypes in the next generation are expected to follow the proportions,

freq AA : freq Aa : freq aa

= p²: 2pq : q²

[because p and q can vary among populations, these proportions will differ from population to population]

The formulae (p², 2pq, q²) tell you how to predict the proportion of genotypes in the next generation, where p is the proportion of A gametes and q is the proportion of a gametes produced by the entire population. The values of p and q will depend upon the relative frequencies of individuals of genotypes AA, Aa and aa in the population.

Each Aa individual produces 50% A and 50% a gametes, but AA individuals produce only A gametes and aa individuals produce only aa gametes. Therefore, the proportion of A and a gametes in the population as a whole (p and q) will depend upon the proportions of each genotype (AA, Aa and aa) present.

At equilibrium, in the absence of evolutionary forces, we can predict (calculate) the frequency of genotypes in the next generation, if we knew the allele frequencies:

freq AA = p²; freq Aa = 2pq; freq aa = q²

Relative frequencies are proportions, or ratios. So, we know that a population is at equilibrium if the genotypes are at these predicted frequencies.

We do not need to observe the population generation after generation to know whether, at any particular time, it is at equilibrium. But it will continue to be at equilibrium only if no evolutionary forces are acting on the population. Assuming this is so, we can predict the genotype frequencies in the next generation (because we already know p and q). This would be: